Topological Spaces

Abstract

We begin by recording a few items from real analysis (our canonical reference for this material is [Sp1], Chapters 1–3, which should be consulted for details as the need arises) . For any positive integer n, Euclideann-space ℝn = {(x1 , ... , xn) : xi ∊ ℝ. , i = 1, ..., n } is the set of all ordered n-tuples of real numbers with its usual vector space structure (x + y = (x1, . . . , xn)+(y1, . . . , yn) = (x1 + y1, . . . , xn + yn) and ax = a(x1, ... , xn) = (ax1, ... , axn)) and norm (||x|| = ((x1)2+ ... +(xn)2)½). An open rectangle in ℝn is a subset of the form (a1 , b1) × .... × (an, bn), where each (ai, bi), i = 1,...., n, is an open interval in the real line ℝ . If r is a positive real number and p ∊ ℝn, then the open ball of radiusr about p is Ur(p) = { x ∊ ℝn: x - p < r}. A subset U of ℝn is open in ℝn if, for each p ∊ U, there exists an r > 0 such that Ur(p) ∈ U (equivalently, if, for each p ∊ U, there exists an open rectangle R in ℝn with p ∊ R ∈ U). The collection of all open subsets of ℝn has the following properties: (a) The empty set Ø and all of ℝn are both open in ℝn. (b) If { Uα : α ∊ A } is any collection of open sets in ℝn (indexed by some set A), then the union Uα∈AUα is also open in ℝn. (c) If { U1, ... , Uk } is any finite collection of open sets in ℝn, then the intersection U1∩...∩ Uk is also open in ℝn.